Properties

Label 676.1.i.a.147.1
Level $676$
Weight $1$
Character 676.147
Analytic conductor $0.337$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [676,1,Mod(23,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.23");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 676.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.337367948540\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.676.1
Artin image: $S_3\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 147.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 676.147
Dual form 676.1.i.a.23.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.00000i q^{5} -1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.00000i q^{5} -1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(-0.500000 + 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{17} +1.00000i q^{18} +(0.866025 - 0.500000i) q^{20} +(0.500000 - 0.866025i) q^{29} +(0.866025 - 0.500000i) q^{32} +1.00000i q^{34} +(0.500000 - 0.866025i) q^{36} +(-0.866025 - 0.500000i) q^{37} -1.00000 q^{40} +(0.866025 + 0.500000i) q^{41} +(-0.866025 + 0.500000i) q^{45} +(0.500000 - 0.866025i) q^{49} -1.00000 q^{53} +(-0.866025 + 0.500000i) q^{58} +(0.500000 + 0.866025i) q^{61} -1.00000 q^{64} +(0.500000 - 0.866025i) q^{68} +(-0.866025 + 0.500000i) q^{72} +1.00000i q^{73} +(0.500000 + 0.866025i) q^{74} +(0.866025 + 0.500000i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.500000 - 0.866025i) q^{82} +(-0.866025 + 0.500000i) q^{85} +(1.73205 + 1.00000i) q^{89} +1.00000 q^{90} +(1.73205 - 1.00000i) q^{97} +(-0.866025 + 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{9} - 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{29} + 2 q^{36} - 4 q^{40} + 2 q^{49} - 4 q^{53} + 2 q^{61} - 4 q^{64} + 2 q^{68} + 2 q^{74} - 2 q^{81} - 2 q^{82} + 4 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 0.500000i −0.866025 0.500000i
\(3\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(5\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(6\) 0 0
\(7\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 1.00000i 1.00000i
\(9\) −0.500000 0.866025i −0.500000 0.866025i
\(10\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(18\) 1.00000i 1.00000i
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0.866025 0.500000i 0.866025 0.500000i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0.866025 0.500000i 0.866025 0.500000i
\(33\) 0 0
\(34\) 1.00000i 1.00000i
\(35\) 0 0
\(36\) 0.500000 0.866025i 0.500000 0.866025i
\(37\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −1.00000
\(41\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0.500000 0.866025i 0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0.500000 0.866025i 0.500000 0.866025i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(73\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) −0.500000 0.866025i −0.500000 0.866025i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 1.00000 1.00000
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(98\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(99\) 0 0
\(100\) 0 0
\(101\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0 0
\(109\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.00000 1.00000
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(122\) 1.00000i 1.00000i
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 1.00000i
\(126\) 0 0
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(137\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 0 0
\(139\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) −0.866025 0.500000i −0.866025 0.500000i
\(146\) 0.500000 0.866025i 0.500000 0.866025i
\(147\) 0 0
\(148\) 1.00000i 1.00000i
\(149\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.500000 0.866025i −0.500000 0.866025i
\(161\) 0 0
\(162\) 0.866025 0.500000i 0.866025 0.500000i
\(163\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(164\) 1.00000i 1.00000i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1.00000 1.00000
\(171\) 0 0
\(172\) 0 0
\(173\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.00000 1.73205i −1.00000 1.73205i
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) −0.866025 0.500000i −0.866025 0.500000i
\(181\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(194\) −2.00000 −2.00000
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(198\) 0 0
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.866025 0.500000i 0.866025 0.500000i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.500000 0.866025i 0.500000 0.866025i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) −0.500000 0.866025i −0.500000 0.866025i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.00000 1.73205i 1.00000 1.73205i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.00000i 1.00000i
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.866025 0.500000i −0.866025 0.500000i
\(233\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) 1.00000i 1.00000i
\(243\) 0 0
\(244\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(245\) −0.866025 0.500000i −0.866025 0.500000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(251\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.00000 −1.00000
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 1.00000i 1.00000i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 1.00000 1.00000
\(273\) 0 0
\(274\) −1.00000 −1.00000
\(275\) 0 0
\(276\) 0 0
\(277\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(282\) 0 0
\(283\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.866025 0.500000i −0.866025 0.500000i
\(289\) 0 0
\(290\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(291\) 0 0
\(292\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(293\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(297\) 0 0
\(298\) 1.00000 1.00000
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.866025 0.500000i 0.866025 0.500000i
\(306\) 0.866025 0.500000i 0.866025 0.500000i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(314\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(315\) 0 0
\(316\) 0 0
\(317\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000i 1.00000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0.500000 0.866025i 0.500000 0.866025i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 0 0
\(333\) 1.00000i 1.00000i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −0.866025 0.500000i −0.866025 0.500000i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 2.00000i 2.00000i
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000i 2.00000i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(361\) 0.500000 0.866025i 0.500000 0.866025i
\(362\) −0.866025 0.500000i −0.866025 0.500000i
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00000 1.00000
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) 1.00000i 1.00000i
\(370\) 0.866025 0.500000i 0.866025 0.500000i
\(371\) 0 0
\(372\) 0 0
\(373\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(387\) 0 0
\(388\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(389\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.866025 0.500000i −0.866025 0.500000i
\(393\) 0 0
\(394\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.00000 −1.00000
\(405\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(410\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.00000i 1.00000i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 0 0
\(433\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 1.00000 1.73205i 1.00000 1.73205i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(458\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(459\) 0 0
\(460\) 0 0
\(461\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(465\) 0 0
\(466\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1.00000 −1.00000
\(483\) 0 0
\(484\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(485\) −1.00000 1.73205i −1.00000 1.73205i
\(486\) 0 0
\(487\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(488\) 0.866025 0.500000i 0.866025 0.500000i
\(489\) 0 0
\(490\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) −1.00000 −1.00000
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0.866025 0.500000i 0.866025 0.500000i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 0.866025 0.500000i 0.866025 0.500000i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) 0.500000 0.866025i 0.500000 0.866025i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 2.00000i 2.00000i
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.866025 0.500000i −0.866025 0.500000i
\(545\) 2.00000 2.00000
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(549\) 0.500000 0.866025i 0.500000 0.866025i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 1.00000i 1.00000i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.500000 0.866025i 0.500000 0.866025i
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0.866025 0.500000i 0.866025 0.500000i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(577\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 1.00000i 1.00000i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.00000 1.00000
\(585\) 0 0
\(586\) −1.00000 −1.00000
\(587\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.866025 0.500000i 0.866025 0.500000i
\(593\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.866025 0.500000i −0.866025 0.500000i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.866025 0.500000i 0.866025 0.500000i
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −1.00000 −1.00000
\(611\) 0 0
\(612\) −1.00000 −1.00000
\(613\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) −1.73205 1.00000i −1.73205 1.00000i
\(627\) 0 0
\(628\) −0.500000 0.866025i −0.500000 0.866025i
\(629\) 1.00000i 1.00000i
\(630\) 0 0
\(631\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0.500000 0.866025i 0.500000 0.866025i
\(641\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(657\) 0.866025 0.500000i 0.866025 0.500000i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.500000 0.866025i 0.500000 0.866025i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(674\) −0.866025 0.500000i −0.866025 0.500000i
\(675\) 0 0
\(676\) 0 0
\(677\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0 0
\(685\) −0.500000 0.866025i −0.500000 0.866025i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.00000i 1.00000i
\(698\) −1.00000 1.73205i −1.00000 1.73205i
\(699\) 0 0
\(700\) 0 0
\(701\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −0.500000 0.866025i −0.500000 0.866025i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.00000 1.73205i 1.00000 1.73205i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 1.00000i 1.00000i
\(721\) 0 0
\(722\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(723\) 0 0
\(724\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) −0.866025 0.500000i −0.866025 0.500000i
\(731\) 0 0
\(732\) 0 0
\(733\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(739\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(740\) −1.00000 −1.00000
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(746\) 1.00000i 1.00000i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.00000i 1.00000i
\(773\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.00000 1.73205i −1.00000 1.73205i
\(777\) 0 0
\(778\) −0.866025 0.500000i −0.866025 0.500000i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(785\) 1.00000i 1.00000i
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 2.00000i 2.00000i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 2.00000 2.00000
\(795\) 0 0
\(796\) 0 0
\(797\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.00000i 2.00000i
\(802\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(809\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(810\) −0.500000 0.866025i −0.500000 0.866025i
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.00000 1.00000
\(819\) 0 0
\(820\) 1.00000 1.00000
\(821\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(822\) 0 0
\(823\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.00000 −1.00000
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0.500000 0.866025i 0.500000 0.866025i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 1.73205 1.00000i 1.73205 1.00000i
\(866\) 1.00000i 1.00000i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 2.00000 2.00000
\(873\) −1.73205 1.00000i −1.73205 1.00000i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(882\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000 2.00000
\(899\) 0 0
\(900\) 0 0
\(901\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.866025 0.500000i 0.866025 0.500000i
\(905\) 1.00000i 1.00000i
\(906\) 0 0
\(907\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 0 0
\(909\) 1.00000 1.00000
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.500000 0.866025i −0.500000 0.866025i
\(915\) 0 0
\(916\) 1.73205 1.00000i 1.73205 1.00000i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.00000 1.00000
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 1.00000i 1.00000i
\(929\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.00000 1.73205i −1.00000 1.73205i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(954\) 1.00000i 1.00000i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.00000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(965\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0.866025 0.500000i 0.866025 0.500000i
\(969\) 0 0
\(970\) 2.00000i 2.00000i
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.00000 −1.00000
\(977\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000i 1.00000i
\(981\) 1.73205 1.00000i 1.73205 1.00000i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(986\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 676.1.i.a.147.1 4
4.3 odd 2 CM 676.1.i.a.147.1 4
13.2 odd 12 52.1.j.a.3.1 2
13.3 even 3 inner 676.1.i.a.23.2 4
13.4 even 6 676.1.b.a.675.1 2
13.5 odd 4 52.1.j.a.35.1 yes 2
13.6 odd 12 676.1.c.b.339.1 1
13.7 odd 12 676.1.c.a.339.1 1
13.8 odd 4 676.1.j.a.191.1 2
13.9 even 3 676.1.b.a.675.2 2
13.10 even 6 inner 676.1.i.a.23.1 4
13.11 odd 12 676.1.j.a.315.1 2
13.12 even 2 inner 676.1.i.a.147.2 4
39.2 even 12 468.1.br.a.55.1 2
39.5 even 4 468.1.br.a.451.1 2
52.3 odd 6 inner 676.1.i.a.23.2 4
52.7 even 12 676.1.c.a.339.1 1
52.11 even 12 676.1.j.a.315.1 2
52.15 even 12 52.1.j.a.3.1 2
52.19 even 12 676.1.c.b.339.1 1
52.23 odd 6 inner 676.1.i.a.23.1 4
52.31 even 4 52.1.j.a.35.1 yes 2
52.35 odd 6 676.1.b.a.675.2 2
52.43 odd 6 676.1.b.a.675.1 2
52.47 even 4 676.1.j.a.191.1 2
52.51 odd 2 inner 676.1.i.a.147.2 4
65.2 even 12 1300.1.w.a.1199.2 4
65.18 even 4 1300.1.w.a.399.2 4
65.28 even 12 1300.1.w.a.1199.1 4
65.44 odd 4 1300.1.bc.a.451.1 2
65.54 odd 12 1300.1.bc.a.1251.1 2
65.57 even 4 1300.1.w.a.399.1 4
91.2 odd 12 2548.1.q.b.263.1 2
91.5 even 12 2548.1.bi.a.1439.1 2
91.18 odd 12 2548.1.q.b.1647.1 2
91.31 even 12 2548.1.q.a.1647.1 2
91.41 even 12 2548.1.bn.a.1667.1 2
91.44 odd 12 2548.1.bi.b.1439.1 2
91.54 even 12 2548.1.q.a.263.1 2
91.67 odd 12 2548.1.bi.b.471.1 2
91.80 even 12 2548.1.bi.a.471.1 2
91.83 even 4 2548.1.bn.a.295.1 2
104.5 odd 4 832.1.bb.a.191.1 2
104.67 even 12 832.1.bb.a.575.1 2
104.83 even 4 832.1.bb.a.191.1 2
104.93 odd 12 832.1.bb.a.575.1 2
156.83 odd 4 468.1.br.a.451.1 2
156.119 odd 12 468.1.br.a.55.1 2
208.5 odd 4 3328.1.v.b.2687.1 4
208.67 even 12 3328.1.v.b.1407.2 4
208.83 even 4 3328.1.v.b.2687.2 4
208.93 odd 12 3328.1.v.b.1407.2 4
208.109 odd 4 3328.1.v.b.2687.2 4
208.171 even 12 3328.1.v.b.1407.1 4
208.187 even 4 3328.1.v.b.2687.1 4
208.197 odd 12 3328.1.v.b.1407.1 4
260.67 odd 12 1300.1.w.a.1199.2 4
260.83 odd 4 1300.1.w.a.399.2 4
260.119 even 12 1300.1.bc.a.1251.1 2
260.187 odd 4 1300.1.w.a.399.1 4
260.223 odd 12 1300.1.w.a.1199.1 4
260.239 even 4 1300.1.bc.a.451.1 2
364.31 odd 12 2548.1.q.a.1647.1 2
364.67 even 12 2548.1.bi.b.471.1 2
364.83 odd 4 2548.1.bn.a.295.1 2
364.135 even 12 2548.1.bi.b.1439.1 2
364.171 odd 12 2548.1.bi.a.471.1 2
364.187 odd 12 2548.1.bi.a.1439.1 2
364.223 odd 12 2548.1.bn.a.1667.1 2
364.275 even 12 2548.1.q.b.263.1 2
364.291 even 12 2548.1.q.b.1647.1 2
364.327 odd 12 2548.1.q.a.263.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
52.1.j.a.3.1 2 13.2 odd 12
52.1.j.a.3.1 2 52.15 even 12
52.1.j.a.35.1 yes 2 13.5 odd 4
52.1.j.a.35.1 yes 2 52.31 even 4
468.1.br.a.55.1 2 39.2 even 12
468.1.br.a.55.1 2 156.119 odd 12
468.1.br.a.451.1 2 39.5 even 4
468.1.br.a.451.1 2 156.83 odd 4
676.1.b.a.675.1 2 13.4 even 6
676.1.b.a.675.1 2 52.43 odd 6
676.1.b.a.675.2 2 13.9 even 3
676.1.b.a.675.2 2 52.35 odd 6
676.1.c.a.339.1 1 13.7 odd 12
676.1.c.a.339.1 1 52.7 even 12
676.1.c.b.339.1 1 13.6 odd 12
676.1.c.b.339.1 1 52.19 even 12
676.1.i.a.23.1 4 13.10 even 6 inner
676.1.i.a.23.1 4 52.23 odd 6 inner
676.1.i.a.23.2 4 13.3 even 3 inner
676.1.i.a.23.2 4 52.3 odd 6 inner
676.1.i.a.147.1 4 1.1 even 1 trivial
676.1.i.a.147.1 4 4.3 odd 2 CM
676.1.i.a.147.2 4 13.12 even 2 inner
676.1.i.a.147.2 4 52.51 odd 2 inner
676.1.j.a.191.1 2 13.8 odd 4
676.1.j.a.191.1 2 52.47 even 4
676.1.j.a.315.1 2 13.11 odd 12
676.1.j.a.315.1 2 52.11 even 12
832.1.bb.a.191.1 2 104.5 odd 4
832.1.bb.a.191.1 2 104.83 even 4
832.1.bb.a.575.1 2 104.67 even 12
832.1.bb.a.575.1 2 104.93 odd 12
1300.1.w.a.399.1 4 65.57 even 4
1300.1.w.a.399.1 4 260.187 odd 4
1300.1.w.a.399.2 4 65.18 even 4
1300.1.w.a.399.2 4 260.83 odd 4
1300.1.w.a.1199.1 4 65.28 even 12
1300.1.w.a.1199.1 4 260.223 odd 12
1300.1.w.a.1199.2 4 65.2 even 12
1300.1.w.a.1199.2 4 260.67 odd 12
1300.1.bc.a.451.1 2 65.44 odd 4
1300.1.bc.a.451.1 2 260.239 even 4
1300.1.bc.a.1251.1 2 65.54 odd 12
1300.1.bc.a.1251.1 2 260.119 even 12
2548.1.q.a.263.1 2 91.54 even 12
2548.1.q.a.263.1 2 364.327 odd 12
2548.1.q.a.1647.1 2 91.31 even 12
2548.1.q.a.1647.1 2 364.31 odd 12
2548.1.q.b.263.1 2 91.2 odd 12
2548.1.q.b.263.1 2 364.275 even 12
2548.1.q.b.1647.1 2 91.18 odd 12
2548.1.q.b.1647.1 2 364.291 even 12
2548.1.bi.a.471.1 2 91.80 even 12
2548.1.bi.a.471.1 2 364.171 odd 12
2548.1.bi.a.1439.1 2 91.5 even 12
2548.1.bi.a.1439.1 2 364.187 odd 12
2548.1.bi.b.471.1 2 91.67 odd 12
2548.1.bi.b.471.1 2 364.67 even 12
2548.1.bi.b.1439.1 2 91.44 odd 12
2548.1.bi.b.1439.1 2 364.135 even 12
2548.1.bn.a.295.1 2 91.83 even 4
2548.1.bn.a.295.1 2 364.83 odd 4
2548.1.bn.a.1667.1 2 91.41 even 12
2548.1.bn.a.1667.1 2 364.223 odd 12
3328.1.v.b.1407.1 4 208.171 even 12
3328.1.v.b.1407.1 4 208.197 odd 12
3328.1.v.b.1407.2 4 208.67 even 12
3328.1.v.b.1407.2 4 208.93 odd 12
3328.1.v.b.2687.1 4 208.5 odd 4
3328.1.v.b.2687.1 4 208.187 even 4
3328.1.v.b.2687.2 4 208.83 even 4
3328.1.v.b.2687.2 4 208.109 odd 4